In this chapter, we study a matrix product that is much simpler than the conventional product, but is much less widely understood. It arises naturally in a variety of ways and enjoys considerable rich structure.

**5.0.1 Definition.** The *Hadamard product* of *A* = [*a*_{ij}] ϵ *M*_{m,n} and *B* = [*b*_{ij}] ϵ *M*_{m,n} is defined by *A*∘*B* ≡ [*a*_{ij}*b*_{ij}] ϵ *M*_{m,n}.

The conformability condition for the Hadamard product is simply that both matrices have the same dimensions, and the multiplication is carried out entry by entry, as in conventional matrix addition. Simply imagine the superposition of one array upon the other with a single scalar multiplication performed in each cell.

then

The Hadamard ...

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